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Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

This paper establishes that changing the initial state in Krylov complexity corresponds to a Christoffel transform of the underlying spectral measure, providing a unified framework to analyze state reorganization, amplitude jumps, and complexity finiteness across various quantum systems through orthogonal polynomial recurrences and kernel projections.

Original authors: Abhishek Chowdhury, Ajit Prasad Mahapatra

Published 2026-07-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Abhishek Chowdhury, Ajit Prasad Mahapatra

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a complex dance performance. The music is the "Hamiltonian" (the rules of the universe that never change), and the dancers are the "quantum states" (the positions and energies of particles). In the world of quantum physics, scientists often want to know how complicated a dance gets as it unfolds. They use a tool called "Krylov complexity" to measure this. Think of it like tracking how far a dancer spreads out across the stage over time. If they stay in one spot, the complexity is low; if they run to every corner, the complexity is high.

Usually, to understand this dance, you have to pick a specific starting dancer (the "initial state") and watch how they move. But what if you want to see how the dance changes if you start with a different dancer, or a mix of dancers, without changing the music? Traditionally, scientists thought you'd have to restart the whole calculation from scratch for every new starting point. It's like having to re-choreograph the entire show just because you swapped the lead dancer. This paper tackles that exact problem: Can we predict how the dance changes if we tweak the starting position, using the data we already have from the original dancer?

The authors, Abhishek Chowdhury and Ajit Prasad Mahapatra, have found a clever mathematical shortcut. They discovered that if you change the starting dancer by applying a specific type of "polynomial filter" (which is just a fancy way of saying you mix the original dancer with a few of their neighbors in a precise recipe), you don't need to restart the whole show. Instead, you can use a set of "connectors"—like a translation guide—that instantly tells you how the new dancer will move, how their complexity will grow, and where they will end up, all based on the original dancer's data.

Here is the magic trick they uncovered: Changing the starting state is mathematically equivalent to changing the "weight" of the music's notes. Imagine the music has a sheet of paper with dots representing different notes. The original dancer hears the dots with a certain weight. If you switch to a new dancer made from a polynomial recipe, it's as if you simply put a new, transparent sheet over the music notes and re-weighted them (a process the paper calls a "Christoffel transform"). The paper proves that this re-weighting allows you to calculate the new dancer's entire journey using a "finite-band" rule. This means the new dancer's movement at any point depends only on a small, fixed number of the original dancer's steps, not the entire history.

The paper doesn't just guess this; it provides exact formulas and proves them for several specific, solvable models of quantum systems. These include:

The Heisenberg–Weyl/Charlier chain: Think of this as a quantum oscillator (like a spring). The authors show that if you jump from one energy level to another (a "number-state jump"), you can calculate the new complexity exactly. They even proved that for these jumps, the complexity is always finite and never blows up, and it's always at least as high as the complexity of the starting "vacuum" state.
The SU(2)/Krawtchouk chain: This represents a spinning object with a limited number of states (like a top that can only spin in a few ways). Here, the "terminal closure" means the dance has a hard stop. The paper shows how to handle the math when the new dancer might accidentally skip over some of the available steps, effectively deleting parts of the dance floor.
The Tight-binding/Chebyshev chain: This models a particle hopping along a line of atoms. Changing the starting state here is like saying, "What if the particle started at atom #5 instead of atom #0?" The paper shows that the math for this is identical to the polynomial jump, allowing them to predict the particle's spread perfectly.

One of the most exciting findings is that this method works even when the new starting state is a complex mix of many different dancers (a "superposition"). The paper also addresses what happens when the system is finite (has a limited size) versus infinite. In finite systems, the "dance floor" has edges. The authors show that if your new starting recipe accidentally tries to step off the edge or land on a spot that doesn't exist (a "spectral atom deletion"), the math automatically adjusts, reducing the size of the dance floor for that specific new dancer.

The paper also introduces a "parent measure" concept. Imagine you have a whole team of potential starting dancers. Instead of calculating the dance for each one individually, you can create a single "master map" (a matrix-valued measure) that contains all the spectral data for the whole team. From this master map, you can extract the specific dance plan for any single dancer or any mix of them. This is powerful because it separates the "music" (the Hamiltonian) from the "starting position" (the seed), allowing physicists to study how the complexity depends on the preparation without having to re-solve the entire physics problem every time.

Crucially, the authors are careful to note that this is an exact mathematical solution for polynomial changes. If you try to change the starting state in a way that isn't a polynomial (like a complex, non-algebraic filter), this specific shortcut might not apply directly, or it might require an infinite number of steps. However, for the vast class of polynomial jumps—which includes many physically relevant scenarios like jumping between energy levels or shifting a particle's location—the paper provides a complete, exact toolkit.

In summary, this paper solves a long-standing puzzle in quantum complexity: How do we update our understanding of a quantum system's evolution when we change the starting point, without doing all the hard work again? The answer is a set of elegant mathematical "connectors" and "projections" that act like a universal translator. They take the known data of one state and instantly generate the full complexity profile for a whole family of related states. This allows scientists to explore how different preparations affect quantum chaos and information spreading, keeping the heavy lifting of the calculation done just once, while the rest follows a predictable, beautiful pattern.

Technical Summary: Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

Problem Statement
State Krylov (or spread) complexity characterizes the unitary evolution of a quantum state within the cyclic subspace generated by a Hamiltonian $H$ and an initial seed state $|K_0\rangle$ . This construction yields a specific ordered Krylov basis $\{|K_n\rangle\}$ and a corresponding Jacobi chain. A fundamental question arises: if the Hamiltonian $H$ is held fixed but the initial state is changed to a new seed $|\psi\rangle$ within the same cyclic subspace, can the resulting Krylov dynamics be recovered from the original Lanczos problem, or must the entire construction be restarted?

While two orthonormal bases of a subspace are related by a unitary transformation, the specific Krylov ordering is intrinsic to the seed's orbit under $H$ and is not preserved by generic basis changes. Consequently, reorthogonalization is typically required even for states within the reference Krylov span. This paper addresses the "fixed- $H$ " problem for a specific class of state changes: polynomial initial-state jumps, where the new seed is a polynomial function of the Hamiltonian acting on the reference seed, $|\psi_Q\rangle \propto Q(H)|K_0\rangle$ .

Methodology
The authors develop an exact relative calculus that maps the dynamics of a polynomially modified seed to the pre-computed reference data without re-running the Lanczos algorithm. The methodology relies on the spectral representation of the problem:

Christoffel Reweighting: Changing the seed from $|K_0\rangle$ to $Q(H)|K_0\rangle$ corresponds to a positive modification of the scalar spectral measure $d\mu(E)$ to a new measure $d\nu_Q(E) = |Q(E)|^2 d\mu(E) / N_Q$ . This is identified as a Christoffel transform of the reference measure.
Finite-Band Transfer: The authors derive exact relations connecting the reference monic orthogonal polynomials $P_n$ $P_{n}$ and the shifted polynomials $R^{[Q]}_n$ $R_{n}^{[Q]}$ . Orthogonality constraints imply that the product $W_Q(E)R^{[Q]}_n(E)$ $W_{Q} (E) R_{n}^{[Q]} (E)$ (where $W_Q = |Q|^2/N_Q$ $W_{Q} = ∣ Q ∣^{2} / N_{Q}$ ) expands into a finite window of reference polynomials $P_m$ $P_{m}$ with $n \le m \le n + \deg(W_Q)$ $n \leq m \leq n + de g (W_{Q})$ .
- This yields connector coefficients $\Gamma^{[Q]}_{n,m}$ that map reference Fourier-orthogonal-polynomial moments $I_m(t)$ directly to the shifted Krylov amplitudes $\tilde{\phi}^{[Q]}_n(t)$ .
- The shifted Jacobi coefficients ( $\tilde{a}_n, \tilde{b}_n$ ) are recovered algebraically from these connectors.
Projected Kernels for Complexity: To compute spread complexity $K_Q(t)$ , which requires summing over all Krylov levels, the authors utilize finite-rank projections of the Christoffel–Darboux kernel. Instead of constructing every shifted Krylov vector, the cumulative probability $\Phi^{[Q]}_\ell(t)$ is obtained by projecting the reference evolution onto the subspace defined by the divisibility constraints of the polynomial $Q$ .
Handling Finite Dimensions and Singularities: The framework explicitly handles:
- Terminal Closure: In finite-dimensional systems, polynomial products are reduced modulo the terminal polynomial $T_d$ .
- Support Loss: If $Q$ vanishes at spectral atoms, those atoms are removed from the measure, reducing the cyclic dimension. The formalism uses quotient algebras and "proper lifts" to handle these cases without introducing spurious sites.
- Complex Superpositions: The method accommodates complex coefficients in $Q$ and confluent roots (repeated roots) via jet constraints.

Key Contributions and Results

Exact Relative Dynamics: The paper establishes that for any polynomial seed $Q(H)|K_0\rangle$ , the shifted Krylov amplitudes and spread complexity are determined entirely by the reference Jacobi data (recurrence coefficients, norms, and moments) and the polynomial $Q$ . No new Lanczos recursion is needed in the full Hilbert space.
Finite-Band Connectors: A central result is the finite-band transfer formula (Eq. 1.5 and 3.2), which expresses shifted amplitudes as a linear combination of a finite number of reference moments. The width of this window is fixed by the degree of the seed polynomial.
Complexity Bounds and Finiteness:
- For the Heisenberg–Weyl/Charlier oscillator (infinite discrete spectrum), the authors prove that spread complexity remains finite for any fixed-degree number-state jump, even when the jump deletes spectral atoms (resonant deletion).
- A vacuum lower bound is established: $K_r(\tau) \ge K_0(\tau)$ for number-state jumps, with strict inequality for $r \ge 1$ .
Solvable Chain Applications: The formalism is applied to three canonical solvable families, providing closed-form results:
- Charlier (Heisenberg–Weyl): Demonstrates root-free remainder calculus for arbitrary number-state jumps and large-index asymptotics.
- Krawtchouk (SU(2)): Handles finite spectral support, terminal closure, and support loss via product-Gram factorizations in the terminal quotient algebra. Weyl reflection symmetry is shown to relate complexities of weight states $r$ and $N-r$ .
- Chebyshev (Tight-binding): Interprets polynomial jumps as changes in the initially occupied lattice site, recovering sine-transform lattice dynamics.
Matrix-Valued Parent Measures: The authors generalize the scalar construction to a finite family of seeds. A matrix-valued parent measure $dM(E)$ is introduced, which encodes the spectral interference of all seeds in a subspace. Rank-one compressions of this matrix recover the scalar measures for pure states, while trace contractions recover the energy measures for mixed states. This separates preparation dependence from changes in the generator or Hilbert-space dimension.
Liouville Space Extension: The paper notes that the construction extends to operator seeds in Liouville space ( $Q(\mathcal{L})\mathcal{O}$ ), where the spectral variable is an energy gap rather than an energy, allowing for the study of mixed-state dynamics and thermal reweighting.

Significance and Claims
The paper claims to provide a complete, exact, and non-perturbative solution to the problem of comparing Krylov complexities for different initial states under a fixed Hamiltonian, provided the states are polynomially related.

Separation of Variables: The construction rigorously separates the dependence on the initial state (preparation) from the dependence on the Hamiltonian (generator) and the Hilbert-space dimension. This allows for the study of "initial-state dependence" as a distinct dynamical feature.
Computational Efficiency: By reusing reference data (moments and recurrence coefficients), the method avoids the computational cost of re-running Lanczos for every new seed, which is particularly valuable in many-body systems where Lanczos calculations are expensive.
Theoretical Unification: The work unifies the spectral theory of Jacobi operators, Christoffel transforms, and orthogonal polynomials with the modern framework of Krylov complexity. It demonstrates that the "nonlinear" reorganization of the Krylov basis due to a seed change admits an "exact relative description" via Christoffel reweighting.
Scope: The authors emphasize that while the transfer formulae are exact for any finite degree, the physical interpretation of a "nearby" jump relies on the fixed-low-degree regime. The framework applies to generic Hamiltonians where the reference Jacobi data is known, not just to integrable systems, though the closed-form examples are drawn from solvable chains.

The paper concludes that this relative calculus offers a robust tool for analyzing the sensitivity of quantum complexity to state preparation, distinguishing it from changes in the underlying dynamics or system size.

Technical Summary: Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

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